# Bayes’ Theorem

Imagine that you come home from a party and you are stopped by the police. They ask you to take a drug test and you accept. The test result is positive. You are guilty.

But wait a minute! Is it really that simple?

In Germany about 2.8 million people consume weed on a regular basis, that’s about 3.5% of the population.

Let’s say D is drug addict and consumes weed regularly, ¬D consumes no weed. So the chance by randomly picking a person that he or she is a drug addict is P(D) = 0.035

Because You either take drugs or you don’t the remaining part must be non-drug takers P(¬D) = 1 – P(D) = 0.965.

The accuracy of a drug test is about 92%. So let’s assume that there are 8% false positives and 8% false negatives as well.

The chance that if a person actually takes drugs the test result will be positive is P(+|D) = 0.92 but You also get a positive reuslt when a person doesn’t take drugs in 8% of all cases: P(+|¬D) = 0.08 These are called “False positives”.

When a person doesn’t take drugs the test will be negative in 92% of all cases P(-|¬D) = 0.92.  And of course a test can also be negative even if a person takes drugs P(-|D) = 0.08. these are called “False negatives”. Got it?

What comes next?

Combined Probabilities

Knowing the success and error rates of the test and the relative distribution of drug consumers we can calculate the combined probabilities:

• P(+, D) = 0.035 * 0.92 = 0.0322
• Think: The test is positive AND the person is a drug user
• P(+, ¬D) = 0.965 * 0.08 = 0.0772
• Think: The test is positive AND the person is NOT a drug user
• P(-, D) = 0.035 * 0.08 = 0.0028
• Think: The test is negative AND the person is a drug user
• P(-, ¬D) = 0.965 * 0.92 = 0,8878
• Think: The test is negative AND the person is NOT a drug user

## Bayes Theorem

P(A|B) = P(B|A) * P(A) / P(B)

In our case we are interested in the probability of a person being a drug addict given the test is positive. That means:

P(D | +) = P(+ | D) * P(D) / P(+) = P(+ | D) * P(D) / ( P(+, D) + P(+, ¬D) )

= 0.92 * 0.035 / (0.0322 + 0.0772) = 0.294

The outcome is quite interesting and mildly shocking: The probability that a person tested positively is actually a drug addict is only around 29% or less than one third!

Why is this so counter intuitive, when the test states an accuracy of 92%?  That is the so called base rate fallacy. We have to take into account that only 3.5% of the population actually take drugs.

Drug tests generally produce false-positive results in 5% to 10% of cases and false negatives in 10% to 15% of cases, new research shows.

## 3 Replies to “Bayes’ Theorem”

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